The old lady in the red scarf was standing inches from the front of am American-made M1 Abrams tank of the Egyptian Third Army, right on the edge of Tahrir Square. Its soldiers were paratroops, some in red berets, others in helmets, gun barrels pointed across the square, heavy machine guns mounted on the turrets. "If they fire on the Egyptian people, Mubarak is finished," she said. "And if they don't fire on the Egyptian people, Mubarak is finished." Of such wisdom are Egyptians now possessed.

Shortly before dusk, four F-16 Falcons – again, of course, manufactured by President Barack Obama's country – came screaming over the square, echoes bouncing off the shabby grey buildings and the giant Nasserist block, as the eyes of the tens of thousands of people in the square stared upwards. "They are on our side," the cry went up from the crowds. Somehow, I didn't think so. And those tanks, new to the square, 14 in all that arrived with no slogans painted on them, their soldiers sullen and apprehensive, had not come – as the protesters fondly believed – to protect them.

But then, when I talked to an officer on one of the tanks, he burst out with a smile. "We will never fire on our people – even if we are ordered to do so," he shouted over the roar of his engine. Again, I was not so sure. President Hosni Mubarak – or perhaps we should now say "president" in quotation marks – was at the military headquarters, having appointed his new junta of former military and intelligence officers. The rumour went round the square: the old wolf would try to fight on to the end. Others said it didn't matter. "Can he kill 80 million Egyptians?"

Anti-American sentiment was growing after Mr Obama's continued if tepid support for the Mubarak regime. "No, Obama, not Mubarak," posters read. And Mr Mubarak's face appeared with a Star of David superimposed over his face. Many of the crowd produced stun-gun cartridge cases fired last week with "Made in the USA" stamped on the bottom. And I noticed the lead tank's hull bore markings beginning "MFR" – at this point a soldier with a rifle and bayonet fixed was ordered to arrest me so I ran into the crowd and he retreated – but could "MFR" stand for the US Mobile Force Reserve, which keeps its tanks in Egypt? Was this tank column on loan from the Americans? You don't need to work out what the Egyptians make of all this.

Yet there were extraordinary scenes earlier in the day between protesters and tank crews of another unit (this time, the machines were older American M-60 Pattons of Vietnam vintage), which appeared to be about to protect a unit of water cannons sent to clear the streets. Hundreds of young men overwhelmed one tank, and when a lieutenant in sun glasses began firing into the air, he was pushed back against his armoured vehicle and had to climb on top to avoid the men. Yet the crowd quickly became good natured, posed for pictures on the tank and handed the soldiers fruit and water.
When a long line of troops assembled across the road, a very old, hunch-backed man sought and gained permission to approach them. I followed him as he embraced the lieutenant and kissed him on both cheeks and said: "You are our sons. We are your people." And then he walked down the row of troops and kissed each one and embraced each one and told each one that he was his son. You need a heart of stone not to be moved by such scenes and yesterday was replete with them.

At one point, a group of protesters brought a man they said was a thief – of which Cairo seems full at the moment – and he was trussed up and handed to the soldiers. "You are here to protect us," they chanted. When one of the soldiers hit the man in the face, his officer slapped him. Then the soldier sat down, shaking his head in despair. All day, an Egyptian Mi-25 helicopter – this time a relic of Soviet ordnance – circled the crowds, six rockets in the pods, but did nothing. Later a French-built Gazelle of the Egyptian air force flew low over the crowds, and the people waved at the place and the pilot could be seen waving back.

And all the time Egyptians walked up to foreigners – and a grey-haired Englishman doesn't look very Egyptian – and insisted that a people who had lost their fear could never be reinjected with fear.

"We will never be afraid again," a young woman shouted at me as the jets screamed over again. And a former cop now claiming to be a liaison man between the demonstrators and the army said that "the army will be with us because they know Mubarak must go". Again, I am not so sure.

And the looting and burning go on. The former policeman – who should know – told me that many of the looters are members of a group which belonged to the Mr Mubarak's National Democratic Party, whose previous role had been to bully Egyptians to go to polling stations and vote for their beloved leader. So why, we all wonder now, are these men trying to loot and burn, crimes which are being blamed on all those who demand that Mr Mubarak leave the country? Those demands, incidentally, now include the expulsion of Omar Suleiman, his former top spy, who is Vice-President.

Across Egypt, and on almost every street in Cairo, there are now vigilantes – not Mubarak men, but ordinary civilians who are tired of the semi-official gangs who are robbing their own people at night-time. To get back to my hotel last night, I had to pass through eight checkpoints of men, young and old – one was stooped, with a walking stick in one hand and an old British .303 Lee Enfield rifle in the other – who are now attacking thieves and handing them to the army. But this is no Dad's Army.

In the early hours of yesterday morning, a group of armed men turned up at the Children's Cancer Hospital near the old Roman aqueduct. They wanted to take the medical equipment, but within minutes, local people ran down the road and threatened the men with knives. They retreated at once. Dr Khaled el-Noury, the chief operating officer at the hospital, told me that the armed visitors were disorganised and apparently frightened of being harmed.

They were right. The reception clerk at the children's hospital showed me the kitchen knife he kept on his desk for protection. Further proof of fighting power lay outside the gate where men appeared holding clubs and sticks and pokers. A boy – perhaps eight years old – appeared brandishing an 18-inch butcher's knife, slightly more than half his height. Other men holding knives of equal length came to shake hands with the foreign journalist.

They are no third force. And they believe in the army. Will the soldiers go into the square? And does it matter if Mr Mubarak goes anyway?